# Probably?

I’ve been wanting to write an article entitled “Oatmeal Raisin: The Cookie Nobody Loves.” Unfortunately, although this title captures, I am convinced, a deep truth, I could not find a way to link it to tax law. So instead of describing why, if you leave out lots of plates of different kinds of cookies and come back a little while later there are always more oatmeal raisin cookies left than any other kind, but if you come back an hour later, all the cookies, including the oatmeal raisin cookies, are gone (nobody loves ’em, but they do like ’em), this post will describe the piece I wrote instead: Probably? Understanding Tax Law’s Uncertainty.

As I described in an earlier post, flipping a coin is risky, because while we do not know whether it will come up heads, we do know the probability that it will come up heads (50%). The presidential election is uncertain, because we do not know whether John McCain will be elected president, and we do not know the probability that he will be elected president. A.J. Sutter pointed out in a comment to that post that the distinction between risk and uncertainty (that is, between known probabilities and unknown probabilities) ties into the debate about the correct interpretation of probability statements. As it happens, that debate is precisely Probably?‘s topic.

We might say that the probability that an event will occur is the number of times that event will occur over the long run out of the number of times that it could occur. So when we say that a coin has a 50% chance of coming up heads, we mean that if we flip the coin a lot of times–a million, say–about half of those flips will come up heads. And the more times we flip, the closer the percentage of heads will get to 50%. This is a frequentist interpretation of a probability statement.

But this interpretation doesn’t work if the event we’re talking about is not risky, but is, rather, uncertain. As others have noted, tax law is uncertain–that is, that we do not, and cannot, know the probability that a court will uphold a particular tax position. Tax advisors make these sorts of probability statements all the time, because a taxpayer faces lower penalties if he can get a tax advisor to give an opinion that there is a certain level of probability that the taxpayer’s position will eventually be upheld by a court. But if we don’t and can’t know this probability, what does it mean to say that there is a, say, 90% chance that a particular tax position will be upheld?

It means, I think, that the speaker believes that there is a 90% chance the tax position will be upheld. Or, put another way, the speaker would pay 90 cents to play a game in which he would get a dollar if the position were upheld and get nothing if it were struck down. This is what’s known as a “subjectivist” interpretation of a probability statement.

So, who cares? Well, everyone should care, of course!

The IRS and tax advisors should care, because a subjectivist interpretation of tax probabilities could provide additional support for stringent and much-criticized laws that regulate the substance of tax advisors’ written opinions. If we think of tax probabilities of expressions of belief, we can see that these strict rules may actually help tax advisors arrive at less biased estimates of the chance that a tax position would be upheld by a court.

Lawmakers should care, because incorporating tax law’s uncertainty into economic models suggests that lawmakers should, perhaps counterintuitively, be cautious of reducing tax law’s uncertainty. Empirical work suggests that some taxpayers do not like uncertainty. So the uncertainty associated with whether certain questionable transactions are permitted (aside from any penalties imposed if transactions do turn out to be forbidden) might itself reduce the number of taxpayers who engage in those transactions.

And legal scholars who are working with economic models should care too, because these models can change if we acknowledge that neither taxpayers nor lawmakers have access to a “true” probability that a tax position will be struck down. For example, instead of just looking at the taxpayer’s estimation that his position will be struck down by a court, a subjectivist interpretation requires us to look at the relationship between the taxpayer’s estimation and the lawmaker’s estimation. (This last point–that legal scholars, as opposed to economists or statisticians or philosophers, should care about the subjectivist/frequentist distinction–is particularly interesting to me, but it is, I think, the subject for a different post.)

### 6 Responses

1. A.J. Sutter says:

1. Thanks for the shout-out. I disagree with one of your fundamental premises, though: I, for one, love oatmeal raisin cookies (as long as they don’t have that sticky brittleness that comes from too much granulated sugar in the recipe). Sorry that I live too far way to free-ride on your cookie events.

3. So does this suggest that the ultra-are-you-thinking-what-she’s-thinking-what-you’re-thinking (the “fog of tax”?) approach of lawmakers should be reserved for big-ticket deductions that might be taken by wealthier taxpayers? BTW, which taxpayers are the ones who are most uncertainty-averse — the wealthier ones or home-office types? If lawmakers should reserve their subtle thinking for the big-ticket tax shelters, but (a) the folks who use shelters are more dice-rolling, or (b) their advisers more prone to look for loopholes in ambiguous rules, does this defeat the enhanced compliance result you desire (or assume that lawmakers desire)? Aren’t most ambiguities likely eventually to get resolved in the course of a cat-and-mouse game between wealthy taxpayers and lawmakers?

2. billb says:

[citation need], given that I love oatmeal raisin cookies.

Also, why is it better to have high uncertainty about a dubious tax proposition than high certainty? If a tax dodge has 99.9% chance of being disallowed by a court, and tax lawyers are 99.9% certain about this, isn’t that more helpful to taxpayers who can then be advised quite accurately not to do it?

Aren’t high levels of uncertainty likely to stop people in their tracks and thus inhibit high numbers of both legit and illegitimate transactions thereby reducing efficiency?

3. Matthew BCL says:

Yeah, not to pile on, but lately my wife’s been making a big deal about how oatmeal cookies are her favorite. I think she likes how they’re more like food than a lot of cookies are – yet they’re still a sweet treat! Perhaps that article should be titled “Oatmeal Cookies: All Things to All People?”

4. AEW says:

I don’t understand how the distinction between risk and uncertainty relates to subjective and frequency interpretations of probability. Say I have two dice, one with dollar signs on 4 sides, and one with dollar signs on two sides. I put one in each hand (behind my back) and let you choose a hand. I take that die and say “I’m going to roll this, and if a dollar sign comes up, you win a dollar.” What are the odds that you win a dollar? You could say that the probability is either 1/3 or 2/3, you don’t know. Or you could say the probability is 0.5. Does this count as uncertainty? It’s true that to say that the probability is 0.5 is subjective in the sense that someone who peeked behind my back would have a different belief of the probability. But it’s fully consistent with a frequentist interpretation of probability in that if we repeated the process ad infinitum, the frequency would approach 50%.

It’s empirically true that people seem to have a psychological aversion to more complicated ways of obtaining an outcome. They seem to prefer a coin flip to the process above. But I don’t see how this is logically related to the philosophical interpretation of probabilities. Am I misunderstanding uncertainty?

5. A.J. Sutter says:

AEW, your dice analogy might be clouding the issue for you. A condition of “risk” requires that you know the chances of an event happening, and frequentists have particular criteria for such knowledge. In theory (though see below), rolling dice is the kind of repeatable event that can meet these criteria. But many events involving human activities aren’t properly analogized to rolling dice, pulling marbles from an urn, watching beta decays of atoms, etc. They don’t occur often enough to know the frequentist, long-run probabilities. Even batting averages don’t make the cut. There might be a good argument to consider one individual’s at-bats as repetitions of an identical event, one of the preconditions for using a frequentist approach. But even in this case, a player’s batting average doesn’t necessarily stabilize the more at bats he or she has; it may have a plateau at some times, but generally it has ups and, especially ultimately, downs.

Suppose you know the resolution of hundreds or thousands of tax shelter cases, and let’s say that the number of cases, had they been dice rolls, might be of a number that we’d agree would be sufficient to establish the frequentist probabilities for dice. Is it really plausible that all these cases, involving different taxpayers, in different tax years, with different judges, counsel, etc., are so similar that your could consider them repetitions of an identical event, for purposes of establishing long-run frequencies of outcomes? Of course, you could *treat* them all as identical events, and compute such a number, but your methodology would be debatable. As I understand Sarah’s point, if they’re not repetitions of an identical event, then the frequentist approach doesn’t apply, which means that you can’t *know* the probabilities of outcomes a priori, which means you aren’t in a “risk” situation.

Actually, even dice rolls can be “uncertain”. In your example, there are a number of assumptions implicit in saying that the frequency would approach 50% if your \$-dice rolls were repeated ad infinitum. For one thing, you might always put the same die into the same hand, and the player might have the strategy of always picking the same hand. For another, in a frequentist world, you’d have to have already rolled those dice an awful lot of times to know a priori that they are fair. If you haven’t done that, then even with a randomized game-playing strategy it’s too early to know what the limiting frequency would be. Of course, maybe you’ve looked at the dice and they seem balanced and symmetrical. But then you don’t really know the probabilities of rolling a ‘\$’ or not; you just assume or trust that, based on their appearance, the dice are fair. (I suppose in theory it might be possible to establish a frequentist probability as to symmetrical and balanced dice being fair, but that would involve collecting data on an awful lot of rolls of an awful lot of dice. You couldn’t rely on statistical inference; since that mathematical framework already assumes the frequentist view, your reasoning would be circular.)